Alexia Walton

Alexia Walton

Introduction

Wastewater (or waste water) is created by human actions ranging from washing dishes to factories releasing water tainted with harmful chemicals into nearby water bodies such as lakes or streams. Humanity typically maintains an “out of sight, out of mind” mentality; the water we use in our homes, yards, schools, and jobs serves to hydrate and clean, then disappears nicely down the drain.

Factory on the Edge of the Water(Craig 2016)

Water Pollution(Ltd. 2017)

But what happens at the end of the drains and pipes? While the specific type of contamination varies depending on the source, generally speaking, used water from “domestic, industrial, commercial, or agricultural activities”, as well as natural occurrences (such as stormwater) is a threat to public health due to the chemicals and substances contaminating the liquid (Wikipedia 2018b). Each year, the health of millions of people is negatively affected due to wastewater finding its way into bodies of water that are considered to be “safe” and are therefore used for activities such as swimming and fishing (Rivers 2017).

Caution Sign in the Myrtle Beach area(Cronin 2014)

In an effort to serve the public good, wastewater treatment facilities are located throughout the United States and internationally (Federation 2018). These facilities serve public health by collecting wastewater, removing the toxic wastes, and making it useable after it has been contaminated due to human actions.

Wastewater Treatment Plant(“Wastewater Treatment Plant” 2016)

Treated water can be reused for other purposes, depending on the initial level of use/contamination, and the process by which it had to be treated. Sometimes the water is released into a water body, returning it directly to the water cycle, but other times it is returned indirectly: used for irrigation purposes, as a non-potable water source, or industrial uses (District, n.d.).

Wastewater Reuse(Waste International 2018)

With the rise of public awareness regarding human impacts on the earth, an increasing amount of attention is being dedicated to developing more economical and efficient methods of wastewater treatment. A new treatment method called anaerobic ammonium oxidation involves treating contaminated liquid in order to remove nitrogen, which causes a variety of environmental and health issues. Excess nitrogen in water can cause aquatic plants and algae to grow in excess and can be harmful when consumed, especially to livestock and infants, as excessive nitrate can result in restricted oxygen transport in the bloodstream (School 2017).

Pollution along Mississippi River(Collaborative 2018)

My Interest in the Data

As an ecologist, I work primarily in the field with plants in order to better understand their physiological processes and how biotic and abiotic factors affect growth, development, spread, and reproduction. Human influences on the environment are major driving factors in the field of ecology. We are witnessing shifts of species due to large scale events such as global warming, as well as smaller-scale events such as deforestation, land management techniques, and industrial developments. As such, understanding exactly how we are impacting the environment, and coming to a better understanding of what we can do to mitigate the effects of our actions is key in becoming better stewards of the environment and helping protect the beautiful nature that exists throughout the world.

The Data

In 2003, researchers collected 120 wastewater samples in order to test the new treatment method. The wastewater was created as a result of being used in an industrial plant which manufactured dry-spun acrylic fibers. The amount of ammonium treatment added was recorded as well as the amount of nitrogen removed, both in milligrams per liter (mg/L) (Mendenhall and Sincich 2016).

Nit=read.csv("NITRO.csv", header=TRUE)
head(Nit)
##   NITRO AMMON
## 1 18.87 67.40
## 2 17.01 12.49
## 3 23.88 61.96
## 4 10.45 15.63
## 5 36.03 83.66
## 6 39.38 69.29

The Problem to be Solved

I have undertaken to analyze the data in order to determine what, if any, statistical relationship exists between the amount of treatment added and the amount of nitrogen removed from wastewater. In an effort to serve the public, the desired outcome of this research and analysis will be to contribute to the development of economic and effective methods of wastewater treatment.

Preliminary Plots and Interpretation of the Data

library (s20x)
pairs20x(Nit)

library(ggplot2)
g = ggplot(Nit, aes(x = AMMON, y = NITRO)) + geom_point()
g = g + geom_smooth(method = "loess")
g

These preliminary plots serve to examine what relationship appears to be present between the data gathered in 2003. It appears to be somewhat uniform and linear, but the following statistical analysis will determine the degree to which this is true, and will determine the extent of the actual relationship.

Theory behind the Analysis

I believe that the value of nitrogen (Y) tends to increase linearly as ammonium (X), that is, when the amount of ammonium added to the wastewater is increased, the amount of nitrogen removed also increases. I want to make a model relating the two variables to one another by drawing a line through all the points. I will define nitrogen as my dependent variable, and ammonium as my independent variable.

I could use a deterministic model if all the data points were perfectly aligned and I didn’t have to worry about errors in my prediction; however, I know from the preliminary graphs that the data points are not perfectly aligned. A probabilistic model will be more accurate, in this instance, as it will take into account the randomness of the distribution of data points around the line. A simple linear regression model (hereafter referred to as SLR) is one type of probabilistic model, and will be used in my data analysis. SLR assumes that the mean value of the y data for any value of the x data will make a straight line when graphed, and that any points which deviate from the line (above or below) are equal to \(\epsilon\). This statement is written as:

\[y = \beta_0 + \beta_1x_i + \varepsilon_i\]

Where \(\beta_0\) and \(\beta_1\) are unknown parameters, \(\beta_0 + \beta_1x\) is the mean value of y for a given x, and \(\varepsilon\) is the random error. Working with the assumption that some points are going to deviate from the line, I know that some will be above the line (positive deviation) and some will be below the line (negative deviation), with an \(E(\varepsilon)=0\). This would make the mean value of y:

\[ \begin{split} E(y) & = E(\beta_0 + \beta_1x_i + \varepsilon_i) \\ & = \beta_0 + \beta_1x_i + E(\varepsilon_i) \\ & = \beta_0 + \beta_1x_i \end{split} \]

Thus, the mean value of y for any given value of x will be represented by \(E(Y|x)\) and wil graph as a straight line, with a y-intercept of \(\beta_0\) and a slope of \(\beta_1\).

In order to fit a SLR model to my dataset I need to first estimate the parameters \(\beta_0\) and \(\beta_1\). In order to make a valid estimate, it will depend on the sampling distributions of the estimators, which depend on the probability distribution of \(\varepsilon\); thus, I need to make the following assumptions about \(\varepsilon\) (Mendenhall and Sincich 2016):

  • the mean of the probability distribution of \(\epsilon\) = 0;
  • the variance of the probability distribution of \(\epsilon\) is constant for all values of the independent variable - for a straight line model this means V(\(\epsilon\)) = a constant for all values of x;
  • the probability distribution of \(\epsilon\) is normal;
  • the errors associated with different observations are independent of one another.

Estimating the Parameters

In order to estimate \(\beta_0\) and \(\beta_1\) I am going to use the method of least squares, that is, I want to determine the line that best fits my data points with a minimum sum of squares of the deviations. This is called the SSE (sum of squares for error). In a straight-line model, I have already discussed that \(y=\beta_0+\beta_1x_i+\varepsilon_i\). The estimator will be \(\hat{y_i}=\hat{\beta_0}+\hat{\beta_1}x_i\). The residual (the deviation of the ith value of y from its predicted value) is calculated by \((y_i-\hat{y_i})=y_i-(\hat{\beta_0}+\hat{\beta_1}x_i)\), thus, the SSE is \(\sum\limits_{i=1}^{n}[y_i-(\hat{\beta_0}+\hat{\beta_1}x_i)]\).

If the model works well with the data we should expect that the residuals are approximately Normal in distribution with mean 0 and constant variance (Mendenhall and Sincich 2016).

Method of Least Squares

nit.lm=lm(NITRO~AMMON,data=Nit)
summary(nit.lm)
## 
## Call:
## lm(formula = NITRO ~ AMMON, data = Nit)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -22.1999  -5.3529   0.4823   4.9763  23.3991 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  2.22255    2.16665   1.026    0.307    
## AMMON        0.57637    0.01757  32.796   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 9.051 on 118 degrees of freedom
## Multiple R-squared:  0.9011, Adjusted R-squared:  0.9003 
## F-statistic:  1076 on 1 and 118 DF,  p-value: < 2.2e-16

This gives us the following values:

\[\hat{\beta_0}=2.22255\]

\[\hat{\beta_1}=0.57637\]

We also see the multiple and adjusted R-squared values, which we will use later in the analysis as we compare models and determine which type of line will be the best fit for the data.

Calculating the Confidence Interval (CI) for Parameter Estimates

ciReg(nit.lm, conf.level=0.95, print.out=TRUE)
##             95 % C.I.lower    95 % C.I.upper
## (Intercept)       -2.06801           6.51311
## AMMON              0.54157           0.61117

The Least-Squares Estimates

\[\hat{\beta_0} + \hat{\beta_1}x_i = 2.22255 + 0.57637*x_i\]

The least-squares estimate of the slope, \(\hat{\beta_1}=0.57637\), indicates that the estimated amount of nitrogen removed from the wastewater increases by 0.58 mg/L for each additional 1.00 mg/L of ammonium added, with this interpretation being valid over the range of ammonium values.

Verifying Assumptions

I want to demonstrate that a straight line is (or is not) the best fit for the data gathered. In order to do this, I need to verify that the assumptions made above are indeed correct.

plot(NITRO~AMMON,bg="Blue",pch=21,cex=1.2,
     ylim=c(0,1.1*max(NITRO)),xlim=c(0,1.1*max(AMMON)),
     main="Scatter Plot and Fitted Line of Nitrogen vs Ammonium", data=Nit)
abline(nit.lm)

A simple scatter plot seems to indicate that a straight line is a good fit for the data, but additional statistics can be run to determine the best fit.

Plot of Residuals

Plotting the residual line segments (the deviations about the fitted line), allows us to visualize the extent to which the points vary from the line we are trying to fit to the data. This also provides us with the RSS (residual sum of squares), which we will use in additional calculations to find the \(R^2\) value of the line we are trying to fit to the dataset.

plot(NITRO~AMMON,bg="Blue",pch=21,cex=1.2,
              ylim=c(0,1.1*max(NITRO)),xlim=c(0,1.1*max(AMMON)),
              main="Residual Line Segments of Nitrogen vs Ammonium", data=Nit)
ht.lm=with(Nit, lm(NITRO~AMMON))
abline(ht.lm)
yhat=with(Nit,predict(ht.lm,data.frame(AMMON)))
with(Nit,{segments(AMMON,NITRO,AMMON,yhat)})
abline(ht.lm)

Plot of Means

We can visualize the difference between the means of the nitrogen data and ammonium data by graphing the mean of nitrogen removed versus the mean of ammonium added with the fitted line and deviations of the fitted line from the mean height added. This will provide us with the MSS (model sum of squares), which we will use in conjunction with the RSS in a few steps.

plot(NITRO~AMMON,bg="Blue",pch=21,cex=1.2,
             ylim=c(0,1.1*max(NITRO)),xlim=c(0,1.1*max(AMMON)),
             main="Mean of Nitrogen vs Ammonium", data=Nit)
abline(nit.lm)
with(Nit, abline(h=mean(NITRO)))
abline(nit.lm)
with(Nit, segments(AMMON,mean(NITRO),AMMON,yhat,col="Red"))

Plot of Means with Total Deviation Line Segments

Plotting the means of nitrogen removed versus ammonium added with the total deviation line segments demonstrates visually \(\hat{y}=\overline{y}\), and provides us with the TSS (total sum of squares).

plot(NITRO~AMMON,bg="Blue",pch=21,cex=1.2,
              ylim=c(0,1.1*max(NITRO)),xlim=c(0,1.1*max(AMMON)),
              main="Total Deviation Line Segments of Nitrogen vs Ammonium", data=Nit)
with(Nit,abline(h=mean(NITRO)))
with(Nit, segments(AMMON,NITRO,AMMON,mean(NITRO),col="Green"))

Using RSS, MSS, and TSS Values

RSS=with(Nit,sum((NITRO-yhat)^2))
RSS
## [1] 9665.803
MSS=with(Nit,sum((yhat-mean(NITRO))^2))
MSS
## [1] 88104.05
TSS=with(Nit,sum((NITRO-mean(NITRO))^2))
TSS
## [1] 97769.85

\(R^2\) is equal to \(MSS/TSS\), which means that the value calculated is the value for the trend line. The closer \(R^2\) is to 1, the better the fit of the trend line.

MSS/TSS
## [1] 0.9011372

This value indicates that the trend line is a significant correct fit for the dataset.

Specifying the Probability Distribution

Lowess Smoother Scatter Plot of Nitrogen versus Ammonium

library(s20x)
trendscatter(NITRO~AMMON, f = 0.5, data = Nit, main="Nitrogen vs Ammonium")

The red lines depict the region of error where the line of best fit could be found. Again, we see a generally linear trend, with a reasonably narrow region of error.

Linear Model Object:

nit.lm = with(Nit, lm(NITRO~AMMON))

Find the Residuals:

height.res = residuals(nit.lm)

Find the Fitted Values:

height.fit = fitted(nit.lm)

Plot the Residuals versus Ammonium

plot(Nit$AMMON,height.res, xlab="AMMON",ylab="Residuals",ylim=c(-1.5*max(height.res),1.5*max(height.res)),xlim=c(0,1.6*max(height.fit)), main="Residuals vs Ammonium")

It looks as though the residuals are somewhat symmetrical about the zero on the y-axis, this indicates that there is no significant deviation from the line of best-fit.

Plot the Residuals versus Fitted values

trendscatter(height.res~height.fit, f = 0.5, data = nit.lm, xlab="Fitted Values",ylab="Residuals",ylim=c(-1.1*max(height.res),1.1*max(height.res)),xlim=c(0,1.1*max(height.fit)), main="Residuals vs Fitted Values")

Again, we see uniformity about 0, further supporting the linear model as the model of best fit for the dataset.

Check Normality:

normcheck(nit.lm, shapiro.wilk = TRUE)

The p-value for the Shapiro-Wilk test is 0.509. The null hypothesis in this case would be that the errors are distributed normally:

\[\varepsilon \sim N(0,\sigma^2)\]

The results of the Shapiro-Wilk test indicate that we do not have enough evidence to reject the null hypothesis (as the p-value 0.51 is significantly greater than the standard of comparison, 0.05), leading us to reasonably assume that the data is distributed normally.

Testing Another Model for Comparison/Validation

But is a linear model the only option for us to test? In order to make a conclusion about the best-fit for this data set, I’m going to test the results against a curve (quadratic).

Fit a Quadratic to the Points

\[y_i = \beta_0 + \beta_1x_i + \beta_2x_i^2\]

quad.lm=lm(NITRO~AMMON + I(AMMON^2),data=Nit)

plot(NITRO~AMMON,bg="Blue",pch=21,cex=1.2,
   ylim=c(0,1.1*max(NITRO)),xlim=c(0,1.1*max(AMMON)),main="Scatter Plot and Quadratic of Nitrogen vs Ammonium",data=Nit)
myplot = function(x){quad.lm$coef[1] + quad.lm$coef[2]*x + quad.lm$coef[3]*x^2}
curve(myplot, lwd = 2, add = TRUE)

Fitting a quadratic to the data does not produce a visibly dissimilar result; it looks completely linear, as we have seen prior to this, but further analysis will clarify the results and allow us to make a final decision.

A Vector of Fitted Values:

quad.fit = c(height.fit)

A Plot of the Residuals versus Fitted Values

plot(quad.lm, which = 1)

There is symmetry about the y-axis 0 value; however, there are outliers labeled, and the linear trend continues to be displayed, even when a quadratic model is applied.

Check Normality:

normcheck(quad.lm, shapiro.wilk = TRUE)

The p-value is 0.512. Again, the results of the Shapiro-Wilk test indicate that we do not have enough evidence to reject the null hypothesis, leading us to again assume that the data is distributed normally.

Summarize the Model

summary(quad.lm)
## 
## Call:
## lm(formula = NITRO ~ AMMON + I(AMMON^2), data = Nit)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -22.202  -5.365   0.477   4.972  23.395 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  2.167e+00  3.905e+00   0.555    0.580    
## AMMON        5.776e-01  7.349e-02   7.859 2.09e-12 ***
## I(AMMON^2)  -5.518e-06  3.216e-04  -0.017    0.986    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 9.089 on 117 degrees of freedom
## Multiple R-squared:  0.9011, Adjusted R-squared:  0.8994 
## F-statistic: 533.2 on 2 and 117 DF,  p-value: < 2.2e-16

This gives us the following values:

\[\beta_0=2.167e+00\]

\[\beta_1=5.776e-01\]

\[\beta_2=-5.518e-06\]

As well as the multiple and adjusted R-squared. I can use these values to compare to the same calculations for the linear model to determine which method, linear or quadratic, fits the data best.

Calculating the Confidence Interval (CI) for Parameter Estimates

ciReg(quad.lm, conf.level=0.95, print.out=TRUE)
##             95 % C.I.lower    95 % C.I.upper
## (Intercept)       -5.56644           9.90028
## AMMON              0.43205           0.72314
## I(AMMON^2)        -0.00064           0.00063

The equation of the fitted line

\[\beta_0 + \beta_1x_i + \beta_2x_i^2 = (2.167e+00) + (5.776e-01)(x) + (-5.518e-06)(x^2)\]

Comparison of the Two Models

Making Predictions using the Two Models

I want to try and predict the amount of nitrogen removed, when the ammonium added is 50, 35, and 20 mg/L:

Model 1:

amount = predict(nit.lm, data.frame(AMMON=c(50,35,20)))
amount
##        1        2        3 
## 31.04108 22.39552 13.74996

Model 2:

amount2 = predict(quad.lm, data.frame(AMMON=c(50,35,20)))
amount2
##        1        2        3 
## 31.03285 22.37597 13.71660

The predictions made using the first model (linear) are slightly larger than the predictions made by the second model (quadratic), but they are extremely close to one another. Further comparisons will be necessary to determine which model will be the best fit for the dataset.

Comparing R Values

The multiple \(R^2\) for the linear model is 0.9011; the adjusted \(R^2\) is 0.9003. The multiple \(R^2\) for the quadratic model is 0.9011; the adjusted \(R^2\) is 0.8994.

According to these results, both models uphold the assumptions made. The variability in the response is explained identically by the linear and quadratic models when you examine the multiple \(R^2\). However, when you examine the adjusted \(R^2\), important differences come to light. The adjusted \(R^2\) “compares the explanatory power of regression models that contain different numbers of predictors” (Frost 2013). This value modifies \(R^2\) and adjusts for the number of predictors in the model. The adjusted \(R^2\) increases when the new term improves the model in a manner that is more than by chance, and decreases when the new term improves the model in a manner that is less than by chance (Frost 2013). Thus, we can tell by the decrease in the quadratic model that it is not improved by the new term; making the linear model a better model for analysis of this data.

Using Anova as Confirmation

To confirm this finding we can use anova to compare the two models and make final conclusions.

anova(nit.lm,quad.lm)
## Analysis of Variance Table
## 
## Model 1: NITRO ~ AMMON
## Model 2: NITRO ~ AMMON + I(AMMON^2)
##   Res.Df    RSS Df Sum of Sq     F Pr(>F)
## 1    118 9665.8                          
## 2    117 9665.8  1  0.024318 3e-04 0.9863

According to these results, we can maintain our null hypothesis that \(\beta_2\) (\(\varepsilon\)) is equal to zero due to the very large p-value. This indicates that the linear fit is better than the quadratic line for describing the dataset.

Avoiding Bias by using Cook’s Distance

Before making a final decision, however, it is important to avoid biases in the data analysis caused by outliers skewing the data. Therefore, we can use Cook’s distance to examine outliers and their contribution to the data set. Outliers may distort the accuracy of a regression, so Cook’s distance calculates the effect of removing any given data point – it estimates the overall influence of the point on the least-squares regression analysis being done. If a point has a large Cook’s distance, it is considered to need a closer examination in order to determine whether the data is valid. Cook’s distance can also be used to determine which, if any, of the areas within the data set require additional data points to be gathered and included (Wikipedia 2018a).

cooks20x(nit.lm)

Cooks Distance for this linear model and data tells me that observation numbers 56, 66, and 80 have large enough values (\(\ge\) 0.10) that they are considered significant and need closer examination.

I can make a new model, made from the same linear model, using the same data, but removing the datum with the highest Cooks distance (80) removed:

nit2.lm=lm(NITRO~AMMON, data=Nit[-80,])

summary(nit2.lm)
## 
## Call:
## lm(formula = NITRO ~ AMMON, data = Nit[-80, ])
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -21.7913  -5.1564   0.5863   5.2495  23.7856 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  1.48389    2.14538   0.692    0.491    
## AMMON        0.58127    0.01734  33.516   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 8.87 on 117 degrees of freedom
## Multiple R-squared:  0.9057, Adjusted R-squared:  0.9049 
## F-statistic:  1123 on 1 and 117 DF,  p-value: < 2.2e-16

Comparing this new model with the first model:

summary(nit.lm)
## 
## Call:
## lm(formula = NITRO ~ AMMON)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -22.1999  -5.3529   0.4823   4.9763  23.3991 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  2.22255    2.16665   1.026    0.307    
## AMMON        0.57637    0.01757  32.796   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 9.051 on 118 degrees of freedom
## Multiple R-squared:  0.9011, Adjusted R-squared:  0.9003 
## F-statistic:  1076 on 1 and 118 DF,  p-value: < 2.2e-16

We can see that the residual standard error decreased, the multiple \(R^2\) increased, and the coefficients shifted values. Based on this comparison we can conclude that the second model (removing one observation) is a better fit for the data.

Conclusion

Nitrogen contamination of water is a widespread issue due to the environmental conditions surrounding the collection of water for human uses. Groundwater is typically used as a major source of water for a variety of needs, ranging from garden irrigation to making sure water parks have a significant reservoir for summer fun (School 2017). Groundwater is easily contaminated by surface seepage, introducing harmful chemicals into the wells of water, which are then utilized in ways that can bring people into harms way. Additionally, harmful chemicals directly introduced to water sources through phenomena such as oil spills, factory waste, surface runoff, powerwashing cars and homes, all contribute to introducing foreign substances into the water which we consume and use without too much thought or concern as to where it came from or what it contains.

Developing methods by which we can treat wastewater and make it safe for human uses and environmental use is key as we strive to be better stewards of human health and the environment. Researchers are actively developing means by which this can be done, as evident by the researchers in 2003 who collected data regarding ammonium addition to wastewater and the subsequent nitrogen removal.

The Research Question and Results

Using the data regarding ammonium addition to wastewater contaminated with nitrogen, I set out to determine whether or not there was a statistical relationship between the addition of ammonium and removal of nitrogen, and if there was a relationship, to determine whether it was linear (a direct relationship) or quadratic (a more indirect relationship).

Based on the SLR statistical analysis that I have performed on the data, a linear model is the best fit for the data, as supported by the upheld assumptions for the SLR model, and the demonstration that the data fits a modified linear model with more accuracy than a quadratic model. That is, there is a generally linear relationship between the amount of ammonium added and the amount of nitrogen removed. This knowledge is significant, as it allows us to predict the amount of nitrogen that will be removed based on the amount of ammonium added to the wastewater.

Suggestions Going Forward

This model could potentially be further improved by removing additional outliers (using the results from Cook’s plot), and would most likely be improved by addition of more data. If the researchers had performed their analysis on more than 120 wastewater samples, the result might have been an even better fit for the linear model, and might have perhaps compensated for the outliers/values flagged by Cook’s plot, resulting in a better overall model.

Additionally, there are other factors which should be examined in relationship to this dataset: does the amount of wastewater affect the efficacy of the ammonium treatment? Assuming the sample sizes were small and the tests were performed in test tubes, does the ammonium treatment have a similar efficacy when applied to larger volumes? Could this processing method be applied to large pools of wastewater?

These and many more questions can be asked about this situation, and there is a great opportunity to make a large difference to the global community if we can work to establish environmentally friendly treatment plans to negate mankind’s harmful impacts.

References

Collaborative, Mississippi River. 2018. “How Nitrogen & Phosphorus Pollution Affects the Mississippi River.” http://www.msrivercollab.org/focus-areas/nitrogen-phosphorus-pollution/.

Craig, Christopher. 2016. “Finding a New Way to Tackle Water Pollution.” https://intpolicydigest.org/2016/09/25/finding-new-way-tackle-water-pollution/.

Cronin, John. 2014. “29 Signs of the Times: Making Beach Pollution Routine in South Carolina.” https://earthdesk.blogs.pace.edu/2014/07/06/29-signs-of-the-times/.

District, Southwest Florida Water Management. n.d. “Reclaimed Water.” http://www.swfwmd.state.fl.us/conservation/reclaimed/.

Federation, Water Environment. 2018. “List of Wastewater Treatment Plants.” https://wef.org/resources/for-the-public/public-information/list-of-water-treatment-plants/.

Frost, Jim. 2013. “Multiple Regression Analysis: Use Adjusted R-Squared and Predicted R-Squared to Include the Correct Number of Variables.” http://blog.minitab.com/blog/adventures-in-statistics-2/multiple-regession-analysis-use-adjusted-r-squared-and-predicted-r-squared-to-include-the-correct-number-of-variables.

Ltd., White Planet Technologies Pvt. 2017. “Water Pollution.” http://www.indiacelebrating.com/environmental-issues/water-pollution/.

Mendenhall, William M., and Terry L. Sincich. 2016. Simple Linear Regression. 6th ed. Florida, USA: Taylor & Francis Group, LLC.

Rivers, American. 2017. “How Sewage Pollution Ends up in Rivers.” https://www.americanrivers.org/threats-solutions/clean-water/sewage-pollution/.

School, The USGS Water Science. 2017. “Nitrogen and Water.” https://water.usgs.gov/edu/nitrogen.html.

Waste International, Water &. 2018. “Stepping up Monitoring for the Gcc’s Water Reuse Drive.” http://www.waterworld.com/articles/wwi/print/volume-28/issue-1/regional-spotlight/middle-east-africa/stepping-up-monitoring-for-the-gcc-s.html.

Wikipedia. 2018a. “Cook’s Distance.” https://en.wikipedia.org/wiki/Cook%27s_distance.

———. 2018b. “Wastewater.” https://en.wikipedia.org/wiki/Wastewater.